What is the Difference Between FFT and PSD?

By Samudrapom DamReviewed by Lexie CornerNov 14 2024

Signal processing and analysis are critical in electrical, electronic, and telecommunication engineering. The power spectral density (PSD) and fast Fourier transform (FFT) are key tools for analyzing and measuring a signal’s frequency content.^1,2

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The FFT converts time-domain data into the frequency domain, enabling visualization of frequency variations. Similarly, the PSD quantifies power distribution across different signal frequencies. By normalizing the PSD magnitude to a one-Hertz bandwidth, a consistent power value is maintained across varying bandwidths.^1,2

Understanding FFT

Fourier transform is a key method for analyzing signals in both time and frequency domains. The FFT is a computational algorithm that efficiently computes the discrete Fourier transform (DFT) and its inverse (IDFT) for a sequence.^1,3

The FFT provides a streamlined approach to calculating Fourier coefficients, reducing the computational load significantly, which benefits applications such as digital spectral analysis and filter simulation.^1,3

The FFT algorithm works on a sequence of N digital samples, where N is generally a power of two. It resolves frequencies up to the Nyquist frequency, yielding N/2 unique frequency components. To avoid aliasing, the signal is conditioned using a low-pass filter to remove frequencies above the Nyquist limit before digitization.^1,3

Direct DFT computation for large samples (for example, 512 or 1024 points) can be computationally demanding. The FFT accelerates DFT calculations by identifying and reusing repetitive calculations. For example, the DFT decomposes a signal into sinusoidal components, each with a distinct amplitude, phase, and frequency.^3,4

The FFT reduces the DFT’s computational complexity from O(N²) to O(N log N), making it manageable for larger datasets. By applying FFT, a signal’s frequency spectrum can be efficiently extracted, revealing its frequency components. The FFT algorithm leverages the algebraic properties of the Fourier matrix to handle redundant computations effectively.^3,4

The FFT algorithm also uses the periodicity of sine functions to streamline calculations, organizing the Fourier matrix into sparse matrices, which reduces the number of necessary calculations. By eliminating redundancies, the FFT enables Fourier transforms in a range of applications.^3,4

How the FFT Algorithm Works

The FFT algorithm attempts to quickly solve a number of smaller problems instead of solving one big problem, which is more difficult. For example, an FFT of size 64 is first divided into two sets of 32, which are further divided into four sets of 16.5.⁵

This process continues, with the sets being divided into smaller subsets—8 sets of 8, then 16 sets of 4, and finally 32 sets of 2. Calculating a DFT for a data set of size 2 is straightforward and requires minimal computation. The FFT performs DFTs on these small subsets, and the results from each stage are combined to produce the final output.⁵

This method enables real-time analysis for digital signal processing applications. The primary computations occur in the "combine" phase, where data from two groups are combined through complex multiplication, followed by a “butterfly” operation to handle the complex conjugate. This calculation gives the Fourier transform its symmetrical properties.⁵

In addition to reducing computational complexity, the FFT algorithm provides a comprehensive spectral analysis across the time axis, making it applicable to various fields, including audio and image processing, spectral analysis, and communications.⁶

For instance, FFT is widely used in spectral analysis to decompose a signal into its frequency components. It is also useful for efficiently performing convolutions, a key operation in many signal-processing tasks. Other applications include notch filtering in image processing and audio signal denoising.⁷

Understanding PSD

PSD is a statistical representation that shows how a signal's power is distributed across different frequencies. It provides a real-valued function representing power per unit frequency, which helps in understanding the frequency-based variations in signal strength.^2,9

PSD offers a more intuitive visualization of signal frequency components than complex DFT. It is derived from the DFT by calculating the mean squared amplitude of each frequency component, averaged over N samples.^2,9

The shape of the PSD plot provides insights into signal characteristics. For example, a broader peak suggests that signal power is spread across a range of frequencies, while a narrower peak indicates concentration at a specific frequency.⁸ The peak value on the plot corresponds to frequencies with higher power levels. PSD also reveals the signal's bandwidth—the range of frequencies containing most of the signal's energy. A wider bandwidth reflects a broader distribution of signal energy.⁸

PSD is particularly useful for analyzing stationary signals, as it allows the identification of dominant frequency components, which are essential in many signal processing applications. These frequency components reveal information about the system generating the signal, such as noise in electronic circuits or vibrations in mechanical systems.^10,11

In vibration analysis, PSD helps detect resonant frequencies, while in noise measurement, it identifies noise sources and their frequency properties. For real-valued signals, the PSD is symmetric, and for wide-sense stationary processes, PSD and autocorrelation form a Fourier transform pair.^10,11

PSD finds applications in fields like signal processing, communications, audio processing, and environmental monitoring. In audio processing, PSD is used for noise reduction, and in environmental monitoring, PSD analysis of environmental sounds can help identify different species.⁸

Key Takeaways on FFT and PSD in Signal Processing

FFT and PSD play complementary roles in signal processing. The FFT is a computational tool that transforms time-domain signals into the frequency domain, efficiently revealing frequency components and reducing the complexity of traditional Fourier transforms. In contrast, PSD provides a statistical view of how signal power is distributed across frequencies, offering a real-valued function to quantify power per unit frequency.

While FFT gives a complex frequency spectrum, PSD is useful for identifying dominant frequencies and analyzing power characteristics. Together, these tools are essential for analyzing and interpreting signals in various applications.

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References and Further Reading

1. Cerna, M, Harvey, AF. (2000). The Fundamentals of FFT-based Signal Analysis and Measurement. [Online] National Instruments. Available at: https://www.sjsu.edu/people/burford.furman/docs/me120/FFT_tutorial_NI.pdf  (Accessed on 10 November 2024)
2. Theodoridis, S. (2020). Probability and Stochastic Processes. Machine Learning (Second Edition), 19-65. DOI: 10.1016/B978-0-12-818803-3.00011-8, https://www.sciencedirect.com/science/article/abs/pii/B9780128188033000118
3. Gasmi, A. (2022). What is Fast Fourier Transform? [Online] Research Gate. Available at:  DOI: 10.13140/RG.2.2.28731.49444, https://www.researchgate.net/publication/362419674_What_is_Fast_Fourier_Transform
4. Hu, J., Jia, F., Liu, W. (2023). Application of Fast Fourier Transform. Highlights in Science, Engineering and Technology. DOI: 10.54097/hset.v38i.5888, https://drpress.org/ojs/index.php/HSET/article/view/5888
5. Ideal School. (n.d.). Fourier Transforms and the Fast Fourier Transform (FFT) Algorithm. [Online] Ideal School. Available at: https://idealschool.edu.in/Lecture%20Notes/ETC%20DEPARTMENT-6th/DISCRETE%20FOURIER%20TRANSFOR%20&%20FAST%20FOURIER%20TRANSFORM.pdf (Accessed on 10 November 2024)
6. Autsou, S., Rassõlkin, A., Vaimann, T., Kudelina, K. (2022). Analysis of possible faults and diagnostic methods of the Cartesian industrial robot. Proceedings of the Estonian Academy of Sciences. DOI: 10.3176/proc.2022.3.04, https://www.researchgate.net/publication/362585488_Analysis_of_possible_faults_and_diagnostic_methods_of_the_Cartesian_industrial_robot
7. Po, LM. (n.d.). FFT and Its Applications. [Online] City University of Hong Kong. Available at https://www.ee.cityu.edu.hk/~lmpo/ee4015/pdf/2022_EE4015_L05B_FFT.pdf  (Accessed on 10 November 2024)
8. GeeksforGeeks. (2024. Power Spectral Density. [Online] GeeksforGeeks Available at https://www.geeksforgeeks.org/power-spectral-density/ (Accessed on 10 November 2024)
9. Oppenheim, AV., Verghese, GC (2010). Power Spectral Density. [Online] MIT. Available at https://ocw.mit.edu/courses/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/8075041184d566103ce7c3f69afc5e75_MIT6_011S10_chap10.pdf (Accessed on 10 November 2024)
10. Thomas, F. (2023). Power spectral density and rms-amplitude. [Online] KIT. Available at https://publikationen.bibliothek.kit.edu/1000158425 (Accessed on 10 November 2024)
11. Cusumano, JP. (2005). The Power Spectral Density and the Autocorrelation. [Online] IMFT. Available at https://www.imft.fr/wp-content/uploads/2021/03/psdtheory.pdf  (Accessed on 10 November 2024)

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